Finally, in the literature on DFT (see [1]}), there are weaker forms of integrability where one
replaces the ordinary Lie bracket by the so-called D-bracket (a generalization of the Dorfman bracket).
In particular, in [1]} it is shown that a small deformation is `weakly integrable with respect to \(K\) '
if and only if a certain Maurer-Cartan equation is satisfied, where the operators involved are associated to the Lie algebroid structure.
The notion of weak integrability refers to the involutivity of a subbundle in terms of the canonical D-bracket
\(\llbracket \cdot ,\cdot \rrbracket ^{\rm can}\) , see [1]} for details.
Adapting these concepts to our formulation, the subbundle \(L_{t}\)  is weakly integrable if, for all \(x,y\in \Gamma (L_{t})\) ,
it holds \((\llbracket x,y \rrbracket ^{\rm can})^{AA^{\prime }}=\beta ^{A}_{\varepsilon }\pi ^{A^{\prime }}\)  for some \(\pi ^{A^{\prime }}\) .
Equivalently, this is \(\beta _{\varepsilon A }(\llbracket x,y \rrbracket ^{\rm can})^{AA^{\prime }}=0\) .
By a direct calculation (using the definition of \(\llbracket \cdot ,\cdot \rrbracket ^{\rm can}\)  given in [1]})
one can show that this is always true in our case,
so any small deformation of \(K\)  is weakly integrable with respect to \(K\) .
